Rail road cars in North America commonly employ double axle swiveling trucks known as “three piece trucks” to permit them to roll along a set of rails. The three piece terminology refers to a truck bolster and pair of first and second sideframes. In a three piece truck, the truck bolster extends cross-wise relative to the sideframes, with the ends of the truck bolster protruding through the sideframe windows. Forces are transmitted between the truck bolster and the sideframes by spring groups mounted in spring seats in the sideframes. The sideframes carry forces to the sideframe pedestals. The pedestals seat on bearing adapters, whence forces are carried in turn into the bearings, the axle, the wheels, and finally into the tracks. The three piece truck relies upon a suspension in the form of the spring groups trapped in a “basket” between each of the ends of the truck bolster and its associated sideframe. For wheel load equalization, a three piece truck uses one set of springs, and the side frames pivot about the ends of the truck bolster in a manner like a walking beam. The 1980 Car & Locomotive Cyclopedia states at page 669 that the three piece truck offers “interchangeability, structural reliability and low first cost but does so at the price of mediocre ride quality and high cost in terms of car and track maintenance.”
Ride quality can be judged on a number of different criteria. There is longitudinal ride quality, where, often, the limiting condition is the maximum expected longitudinal acceleration experienced during humping or flat switching, or slack run-in and run-out. There is vertical ride quality, for which vertical force transmission through the suspension is the key determinant. There is lateral ride quality, which relates to the lateral response of the suspension. There are also other phenomena to be considered, such as truck hunting, the ability of the truck to self steer, and, whatever the input perturbation may be, the ability of the truck to damp out undesirable motion. These phenomena tend to be inter-related, and the optimization of a suspension to deal with one phenomenon may yield a system that may not necessarily provide optimal performance in dealing with other phenomena.
In terms of optimizing truck performance, it may generally be desirable to obtain a measure of self steering in the truck, desirable to avoid truck hunting, and desirable to have a relatively soft lateral and vertical response. It would be advantageous to be able to obtain the desirable relatively soft dynamic response to lateral and vertical perturbations, to obtain a measure of self steering, and yet to maintain resistance to lozenging (or parallelogramming). Lozenging, or parallelogramming, is non-square deformation of the truck bolster relative to the side frames of the truck as seen from above. It may also be desirable to obtain a measure of self-steering. Self steering may tend to be desirable since it may reduce drag and may tend to reduce wear to both the wheels and the track, and may give a smoother overall ride.
In general, the lateral stiffness of the suspension may tend to reflect the combined lateral displacement of (a) the sideframe between (i) the bearing adapter and (ii) the bottom spring seat (that is, the sideframes may swing or rock laterally), and (b) the lateral deflection of the springs between (i) the lower spring seat in the sideframe and (ii) the upper spring mounting against the underside of the truck bolster, and (c) the moment and the associated transverse shear force between the (i) spring seat in the sideframe and (ii) the upper spring mounting against the underside of the truck bolster.
In a conventional rail road car truck, the lateral stiffness of the spring groups may sometimes be estimated as being approximately half of the vertical spring stiffness. Thus the choice of vertical spring stiffness may strongly affect the lateral stiffness of the suspension. There is another component of spring stiffness due to the unequal compression of the inside and outside portions of the spring group as the bottom spring seat rotates relative to the upper spring group mount under the bolster.
It may be desirable to have springs of a given vertical stiffness to give certain vertical ride characteristics, and a different characteristic for lateral perturbations. For example, a softer lateral response through the main spring groups may be desired at high speed (greater than about 50 m.p.h.) and relatively low amplitude to address a truck hunting concern, while a different spring characteristic may be desirable to address a low speed (roughly 10-25 m.p.h.) roll characteristic, particularly since the overall suspension system may have a roll mode resonance lying in the low speed regime.
For the purposes of rapid estimation of truck lateral stiffness, the following formula can be used:ktruck=2×[(ksideframe)−1+(kspring shear)−1]−1 
where                ksideframe=[kpendulum+kspring moment]        kspring shear=The lateral spring constant for the spring group in shear.        kpendulum=The force required to deflect the pendulum per unit of deflection, as measured at the center of the bottom spring seat.        kspring moment=The force required to deflect the bottom spring seat per unit of sideways deflection against the twisting moment caused by the unequal compression of the inboard and outboard springs.        
In a pure pendulum, the relationship between weight and deflection is approximately linear for small angles of deflection, such that, by analogy to a spring in which F=kx, a lateral constant (for small angles) can be defined as kpendulum=W/L, where k is the lateral constant, W is the weight, and L is the pendulum length. Further, for the purpose of rapid comparison of the lateral swinging of the sideframes, an approximation for an equivalent pendulum length for small angles of deflection can be defined as Leq=W/kpendulum. In this equation W represents the sprung weight borne by that sideframe, typically ¼ of the total sprung weight for a symmetrical car. For a conventional truck, Leq may be of the order of about 3 or 4 inches. For a swing motion truck, Leq may be of the order of about 10″. As noted above, one of the features of a swing motion truck is that while it may be quite stiff vertically, and while it may be resistant to parallelogram deformation because of the unsprung lateral connection member, namely the transom, frame brace, or lateral reinforcement rods, it may at the same time tend to be laterally relatively soft.
One way to obtain a measure of passive self steering is to mount elastomeric pads between the pedestal seat and the bearing adapter. That is to say, when a conventional truck enters a curve, the leading outer wheel may tend to want to pull ahead relative to the leading inner wheel, and the inner wheel may then tend to want to slip, or skid, somewhat. The converse may tend to occur on the trailing axle. This tendency to slip or skid may be reduced somewhat if the axles are able to steer a bit, and thereby to conform to some extent to the curve. Elastomeric pads, sometimes manufactured by Lord Corp., have sometimes been used for this purpose, and may provide a resilient means for permitting some self steering to take place.
Considering the interface between the pedestal seat and the wheelsets at the bearing adapters, there are, potentially, six degrees of freedom, namely vertical, longitudinal and transverse translation, and rotation about each of the vertical, longitudinal, and lateral axes. For the purposes of analysis, in the vertical direction the connection can be approximated as being nearly infinitely stiff. In the longitudinal direction, the stiffness with an elastomeric pad is a function of the shear modulus of the elastomer, the area of the elastomer in plan view, and the thickness of the elastomer. If the elastomer is of constant thickness, and is more or less flat, the lateral stiffness may tend to be roughly the same in both longitudinal and lateral shear. The pad may tend to have torsional compliance about the vertical axis to permit the typically relatively small angular deflection of steering.
Longitudinal cylindrical rockers have been employed to increase warp stiffness by compelling the fore and aft bearing adapter interfaces to swing in unison on a common hinge line. Where substantially cylindrical rockers of relatively close radii are used, (that is, where the radius of curvature of the rocker is relatively close to the radius of curvature of the seat) as for example in U.S. Pat. No. 5,544,591 of Armand Taillon, issued Aug. 13, 1996, the torsional stiffness about the vertical, or z, axis of the interface between the bearing adapter crown and the pedestal seat roof may be very high, such that it may tend to provide resistance to unsquaring relative movement between the wheelsets and side frames.